Problem:
How many positive three-digit integers have a remainder of when divided by , a remainder of when divided by , and a remainder of when divided by
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Suppose is a number that satisfies these conditions. We know that . The first statement implies that or . This, in turn, implies that .
Similarly, the second statement implies that or . This, in turn, implies that .
Finally, the third statement implies that or . This, in turn, implies that .
Together, these three conditions mean that , , and . Thus, we know that . Therefore, .
We also know , so we can see that there are 5 possible values in this interval such that .
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions